------------------------------------------------------------------------
-- The Agda standard library
--
-- Indexed monads
------------------------------------------------------------------------

-- Note that currently the monad laws are not included here.

module Category.Monad.Indexed where

open import Category.Applicative.Indexed
open import Function
open import Level

record RawIMonad {i f} {I : Set i} (M : IFun I f) :
                 Set (i  suc f) where
  infixl 1 _>>=_ _>>_ _>=>_
  infixr 1 _=<<_ _<=<_

  field
    return :  {i A}  A  M i i A
    _>>=_  :  {i j k A B}  M i j A  (A  M j k B)  M i k B

  _>>_ :  {i j k A B}  M i j A  M j k B  M i k B
  m₁ >> m₂ = m₁ >>= λ _  m₂

  _=<<_ :  {i j k A B}  (A  M j k B)  M i j A  M i k B
  f =<< c = c >>= f

  _>=>_ :  {i j k a} {A : Set a} {B C} 
          (A  M i j B)  (B  M j k C)  (A  M i k C)
  f >=> g = _=<<_ g  f

  _<=<_ :  {i j k B C a} {A : Set a} 
          (B  M j k C)  (A  M i j B)  (A  M i k C)
  g <=< f = f >=> g

  join :  {i j k A}  M i j (M j k A)  M i k A
  join m = m >>= id

  rawIApplicative : RawIApplicative M
  rawIApplicative = record
    { pure = return
    ; _⊛_  = λ f x  f >>= λ f'  x >>= λ x'  return (f' x')
    }

  open RawIApplicative rawIApplicative public

record RawIMonadZero {i f} {I : Set i} (M : IFun I f) :
                     Set (i  suc f) where
  field
    monad : RawIMonad M
         :  {i j A}  M i j A

  open RawIMonad monad public

record RawIMonadPlus {i f} {I : Set i} (M : IFun I f) :
                     Set (i  suc f) where
  infixr 3 _∣_
  field
    monadZero : RawIMonadZero M
    _∣_       :  {i j A}  M i j A  M i j A  M i j A

  open RawIMonadZero monadZero public